Abstract

In this article, we investigate the global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system. Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C1 solutions with the bounded L1∩ L∞ norm of the boundary data as well as their derivatives. Based on the existence result, we can prove that when t tends to in nity, the solutions approach a combination of piece-wised C1 traveling wave solutions. As the important example, we apply the results to the chaplygin gas system.

Keywords

1 Introduction and main results

For the general first order quasilinear hyperbolic systems,

∂u∂t+A(u)∂u∂x=0

the global existence of classical solutions of Cauchy problem has been established for lin-early degenerate characteristics or weakly linearly degenerate characteristics with various smallness assumptions on the initial data by Bressan [1], Li [2], Li and Zhou [3, 4], Li and Peng [5, 6], and Zhou [7]. The asymptotic behavior has been obtained by Kong and Yang [8], Dai and Kong [9, 10]. For linearly degenerate diagonalizable quasilinear hyperbolic systems with "large" initial data, asymptotic behavior of the global classical solutions has been ob-tained by Liu and Zhou [11]. For the initial-boundary value problem in the first quadrant Li and Wang [12] proved the global existence of classical solutions for weakly linearly degenerate positive eigenvalues with small and decay initial and boundary data. The asymptotic behavior of the global classical solutions is studied by Zhang [13]. The global existence and asymptotic behavior of classical solutions of the initial-boundary value problem of diagonal-izable quasilinear hyperbolic systems in the first quadrat was obtained in [14].

However, relatively little is known for the Goursat problem with characteristic boundaries. Global existence of the global classical solutions for the Goursat problem of reducible quasilinear hyperbolic system was obtained in [15]. Under the assumptions of boundary data is small and decaying, the global existence and asymptotic behavior to classical solutions can be obtained by Liu [16, 17]. The asymptotic behavior of classical solutions of Goursat problem for reducible quasilinear hyperbolic system was shown in [18].

In this article, we consider the following diagonalizable quasilinear hyperbolic system:

∂ui∂t+λi(u)∂ui∂x=0

(1)

where u = (u1, ..., un)T is unknown vector-valued function of (t, x). λi(u) is given by C2 vector-valued function of u and is linearly degenerate, i.e.,

∂λi(u)∂ui≡0

(2)

The system (1) is strictly hyperbolic, i.e.,

λ1(u)<⋯<λm(u)<0<λm+1(u)<⋯<λn(u)

(3)

Suppose that there exists a positive constant δ such that

λi+1(u)-λi(v)≥δ,i=1,…,n-1

(4)

Consider the Goursat problem for the strictly quasilinear hyperbolic system (1), in which the solutions to system (1) is asked to satisfy the following characteristic boundary conditions:

x=x1(t):u=ϕ(t)

(5)

and

x=xn(t):u=ψ(t);

(6)

where x = x1(t) and x = xn(t) are the leftmost and the rightmost characteristics passing through the origin (t, x) = (0, 0), respectively, such that

dx1(t)dt=λ1ϕ(t)x1(0)=0

(7)

and

dxn(t)dt=λnψ(t)xn(0)=0

(8)

moreover,

l1(ϕ(t))ϕ′(t)≡0

(9)

and

ln(ψ(t))ψ′(t)≡0

(10)

where ϕ (t) = (ϕ1(t), ..., ϕn(t))T and ψ(t) = (ψ1(t), ..., ψn(t))T are any given C1 vector functions satisfying the conditions of C1 compatibility at the origin (0, 0):

ϕ(0)=ψ(0)

(11)

and

λn(ϕ(0))ϕ′(0)-λ1(ψ(0))ψ′(0)+A(ϕ(0))(ψ′(0)-ϕ′(0))=0.

(12)

li(u) be a left eigenvector corresponding to λi(u) and A(u) = diag{λ1(u), ..., λn(u)}.

Our goal in this article is to get the global existence and asymptotic behavior of the global classical solutions of the Goursat problem (1), (5), and (6) with "large" boundary data. With the assumptions that

Remark 1.1. If the system (1) is non-strictly hyperbolic but each characteristic has constant multiplicity, then the result is similar as Theorems above.

2 Global existence of C1solutions

In this section, we will obtain some uniform a priori estimate which also play an important role in the proof of Theorem 1.1. In order to proving the global existence of classical solutions of the Goursat problem (1), (5), and (6), we will prove that ||u||C1(D) is bounded. For any fixed T ≥ 0, we denote DT = {(t, x)| 0 ≤ t ≤ T, x1(t) ≤ x ≤ xn(t)} and introduce

wi(t,x)=∂ui(t,x)∂xi=1,…,n,W1(T)=sup0≤t≤T∫x1(t)xn(t)|w(t,x)|dx

(16)

W̃1(T)=maxi≠jsupC̃j∫C̃j|wi(t,x)|dt,Ũ1(T)=maxi≠jsupC̃j∫C̃j|ui(t,x)|dt

(17)

W̄1(T)=maxi≠jsupLj∫Lj|wi(t,x)|dt,Ū1(T)=maxi≠jsupLj∫Lj|ui(t,x)|dt

(18)

W∞(T)=sup(t,x)∈DT|w(t,x),U∞(T)=sup(t,x)∈DT|u(t,x)

(19)

where C̃j stands for any given j th characteristic dxdt=λj(u), Lj stands for any given radial that has the slope λj(0) on the domain DT .

Lemma 2.1. Under the assumptions of Theorem 1.1, there exists a positive constant C such that, the following estimates hold

W̃1(T),W̄1(T),W1(T)≤CN2

(20)

Ũ1(T),Ū1(T)≤CN1eCN2

(21)

W∞(T)≤CMeCN2

(22)

U∞(T)≤C

(23)

Remark 2.1. The positive constant C is only depend on δ, M0 and independent of M, N1, N2, T. In the following, the meaning of C is similar but may change from line to line.

Proof. For any fixed point (t, x) ∈DT , we draw the i th characteristic C̃i:x=xi(t) through this point and intersecting x1(t) or xn(t) at a point (t*, x1(t*)) or (t*, xn(t*)). Noting system (1), ui(t, x) is a constant along the i th characteristic, then we have ui(t, x) = ϕi(t*) or ui(t, x) = ψi(t*). Then

Case 1. For any fixed t0∈R+, let C̃j:x=xj(t),j>1 stands for any given j th characteristic, passing through any point A(t0, x1(t0)) on the boundary x = x1(t) and intersects t = T at point P. We draw an i th characteristic C̃i:x=xi(t) from P downward, intersecting x = x1(t) at a point B(t1, x1(t1)). Without loss of generality, we assume t0< t1, then j > i. Integrating (26) in the region APB to get

∫C̃j(λj(u)-λi(u))|wi(t,x)|dt=∫t0t1(λi(u)-λi(u)|wi(t,x1(t))|dt

(27)

Along the 1th characteristic, dxdt=λ1(u), then dtdx=1λ1(u)

∂ui(t,x1(t))∂x=∂ui(t,x1(t))∂t∂t∂x=ϕi′(t)λ1(u)

Noting (4), (13), and (23), we can get

∫C̃j|wi(t,x)|dt≤C∫0∞|ϕi′(t)|dt≤CN2

(28)

Case 2. For any fixed t0∈R+, passing through the point A(t0, x1(t0)), we draw C̃j:x=xj(t) and intersecting t = T at point P. We draw the i th characteristic C̃i:x=xi(t) from P downward, intersecting x = xn(t) at B(t1, xn(t1)). Then, we integrate (26) in the region PAOB to get

Combining (24), (30), (31), (34), (37), (43), and (44) together we can obtain the conclusion of Lemma 2.1.

Proof of Theorem 1.1.

Noting the conclusion of Lemma 2.1, we can get

W̃1(∞),W̄1(∞),W1(∞)≤CN2

(45)

Ũ1(∞),Ū1(∞),≤CN1eCN2

(46)

W∞(∞)≤CMeCN2

(47)

U∞(∞)≤C

(48)

Therefore, we can obtain that the system (1), (5), and (6) have global classical solutions on the domain D = {(t, x)| t ≥ 0, x1(t) ≤ x ≤ xn(t)}.

3 Asymptotic behavior of global classical solutions

In this section, under the assumption of the leftmost and rightmost characteristics, we will study the asymptotic behavior of the global classical solutions of system (1), (5), and (6) and give the proof of Theorem 1.2.

Let

DDit=∂∂t+λi(0)∂∂x

(49)

Obviously,

DDit=ddit+λi(0)-λi(u)∂∂x

where ddit=∂∂t+λi(u)∂∂x. Thus, noting system (1)

DuiDit=λi(0)-λi(u)∂ui∂x

(50)

Using the Hadamard's Lemma, we can obtain

DuiDit=-∑j≠i{Λij(u)ujwi}

(51)

where Λij(u)=∫01∂λi(su1,…,sui-1,ui,sui+1,…,sun)∂ujds.

For any fixed point (t, x) ∈D, Passing through (t, x), we draw down the characteristic x = xr(t), which intersect with the characteristic boundary in the point (xr-1(α),α). Then α=x-λi(0)(t-xr-1(α)) (where r = 1, when i = m + 1, ..., n or r = n, when i = 1, ..., m).

In what follows, we will study the regularity of Φ(α). Noting Equation (52), we can get Φ(α) ∈C0 (R). From any fixed point A(t,x)=(t,a+λi(0)(t-xr-1(α))), we draw a characteristic C̃j intersecting the boundary x = x1(t) or x = xn(t) at (xr-1(θi(t,α)),θi(t,α)).

Then, integrating it along the i th characteristic, we obtain

α+λi(0)t-xr-1(α)=θ(t,α)+∫xr-1(θi(t,α))tλiu(τ,xi(τ,θi(t,α)))dτ

(55)

Then, we can get the following Lemma

Lemma 3.1 Under the assumptions of Theorem 1.1, for the θi(t, α) defined above, there exists a unique ϑi(α), such that

limt→+∞θi(t,α)=ϑi(α).

(56)

Proof. Using the Hardarmad's formula, we can rewrite (55) as following

where r is either 1 or n. Then, we know that when t tends to ∞, the right hand of (58) convergence absolutely. For any given α, the right hand of (57) convergence to some function with respect to α. That implies that there exists a unique function ϑ(α), such that

limt→+∞θi(t,α)=ϑi(α)

(59)

Lemma 3.2 Under the assumptions of Theorem 1.1, for any given point (xr-1(α),α) on the boundary, there exists a unique function Ψi(ϑ(α)) ∈C0, such that

limt→+∞wit,α+λi(0)(t-xr-1(α))=Ψi(ϑi(α)).

(60)

uniformly for any α ∈ R.

Proof. Noting

wi(t,α+λi0(t-xr-1(α)))=wi(t,xi(t,θi(t,α))).

(61)

In the following, we prove that there exists a unique Ψi(ϑ(α)) ∈C0, such that

Using the similar procedure, when α > 0, i.e., i = s + 1, ..., n and r = n, we can get

dΦiαdα=1-λi0λnxn-1α,αΨiϑiα.

(68)

Noting the above lemmas, we get the conclusion of Theorem 1.2.

4 Applications

Recent years observations of the luminosity of type Ia distant supernovae point towards an accelerated expansion of the universe, which implies that the pressure p and the energy density ρ of the universe should violate the strong energy condition, i.e., ρ + 3p < 0. Here, we consider a recently proposed class of simple cosmological models based on the use of peculiar perfect fluids [19]. In the simplest case, we study the model of a universe filled with the so called Chaplygin gas, which is a perfect fluid characterized by the following equation of state p=-Aρ=-Aτ, where A is a positive constant.

In Lagrange coordinate, the 1D gas dynamics equations in isentropic case can be written as

τt-ux=0ut+p(τ)x=0

(69)

Noting (69), in isentropic case we can get the system of one dimensional Chaplygin gas model

τt-ux=0ut-Aτx=0

(70)

Nothing systems (70) is linear systems, it is easy to get the eigenvalues

λ+=A,λ-=-A.

(71)

and left eigenvectors

l+=A,1,l-=-A,1

(72)

Introduce the Riemann invariants

v1=u+Aτ,v2=u-Aτ

(73)

we can rewrite the system as following

∂v1∂t-A∂v1∂x=0∂v2∂t-A∂v2∂x=0

(74)

Consider the Goursat problem for system (69) with following characteristic boundary conditions:

x=-At:τ=τ-t,u=u-t;x=At:τ=τ+t,u=u+t

(75)

Then, the above system satisfies the assumptions of Theorems 1.1 and 1.2. More precisely, we can get the following theorems:

Declarations

Acknowledgements

The first author was supported by NSFC-Tianyuan Special Foundation (No. 11126058), Excellent Young Teachers Program of Shanghai and the Shanghai Leading Academic Discipline Project (No. J50101). The second author was supported by China Postdoctoral Science Foundation (No. 2011M501295) and the Fundamental Research Funds for the Central Universities (No. 2011QNZT102). The authors would like to thank Professor Zhou Yi for his guidance and encouragements.

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